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The Poincaré inequality is an open ended condition. (English) Zbl 1180.46025
A function $$u$$ satisfies a $$(1,p)$$-Poincaré inequality, $$p > 1$$, if there exist $$C, \lambda \geq 1$$ such that
$|B(x,r)|^{-1} \int_{B(x,r)} |u - u_{B(x,r)}| \,d\mu\leq Cr \bigg(|B(x,\lambda r)|^{-1} \int_{B(x,\lambda r)} g^p \,d\mu\bigg)^{1/p}$
for every $$x \in X$$ and $$r > 0$$. Here, $$X$$ is a metric space with a doubling Borel measure $$\mu$$, $$g$$ is an upper gradient of $$u$$, and $$u_{B(x,r)}$$ is the mean value of $$u$$ in $$B(x,r)$$. For $$X = \Omega$$ an open set in $$\mathbb{R}^n$$, and $$\mu = m$$ the Lebesgue measure in $$\mathbb{R}^n$$, this reduces to the ordinary Poincaré inequality of Sobolev functions, sometimes called a weak $$(1,p)$$-Poincaré inequality in the case $$\lambda > 1$$, see P. Hajłasz and P. Koskela [“Sobolev met Poincaré” (Mem. Am. Math. Soc. 688) (2000; Zbl 0954.46022)] for various aspects of Poincaré and Sobolev inequalities.
The authors show that, if $$X$$ is a complete metric space, then there exists $$\varepsilon > 0$$ such that a $$(1,q)$$-Poincaré inequality holds for $$q >p-\varepsilon$$ whenever a $$(1,p)$$-Poincaré inequality holds, i.e., that the Poincaré inequality is an open ended property. The result has an important consequence for $$p$$-admissible weights in $$\mathbb{R}^n$$: they display the same open ended property as Muckenhoupt’s $$A_p$$ weights. It also shows that various definitions for Sobolev spaces in metric spaces, due to J. Cheeger [Geom. Funct. Anal. 9, No. 3, 428–517 (1999; Zbl 0942.58018)], P. Hajłasz [Potential Anal. 5, No. 4, 403–415 (1996; Zbl 0859.46022)] and N. Shanmugalingam [Rev. Mat. Iberoam. 16, No. 2, 243–279 (2000; Zbl 0974.46038)], coincide under the above assumptions. The proof makes use of careful estimates of the maximal function of $$u - u_{B(x,r)}$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 43A85 Harmonic analysis on homogeneous spaces
##### Keywords:
Poincaré inequality; metric measure space
##### Citations:
Zbl 0954.46022; Zbl 0942.58018; Zbl 0859.46022; Zbl 0974.46038
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